Obviously connecting in HKG. How does the connection work. Do I need to clear customs and imigration? Reclaim luggage? How much time should I allow? Looks like Priority Lounges, including Chase are available.

How is the premium economy product?

You don't need to clear customs or immigration or reclaim bags - unless you've booked completely separate tickets for each leg (and if those are with the same airline - presumably they are CX based on your question - or even different airlines with an interline agreement you still may be able to combine them for those purposes.)

To address the 'how much time should I allow' part - not that much time at all

Firstly, transferring in HKG doesn't take that much time. The minimum connection time (on this itinerary) is 50mins and is easily doable. If it even get close to that/below that due to delays of the inbound flight, there will be CX staff meeting your flight at the gate upon arrival to help you out. This can either be:
- Escorting you to the gate
- Rebooking you onto a new flight

I would just book the minimum connection as I think Osaka has more to offer than HKG airport

The Premium Economy product is amazing, especially on the JFK-HKG route given the length of the flight (amongst other factors). Elevated service and smaller/quieter/quainter cabin of around 30-40.

Delayed arrival in HK can complicate things even if it meets MCT. I once traveled from PVG-HKG-JFK. It was tight getting into HK and I managed to get on the flight to JFK but not my luggage. It took me a couple of days of waiting before I got back my luggage in the US.